Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 4

Specify the branch before defining a signed jump. Choose $\varphi <\arg z<\varphi +2\pi$, and let $F_{+}$ be the value on the side with argument tending to $\varphi$ from above, while $F_{-}$ is the value on the other side, with argument tending to $\varphi +2\pi$ from below. The entire function $f$ has the same limiting value on both sides, soThus the jump of a fractional power across a branch cut isEquivalently the ratio of the limiting values is $F_{-}/F_{+}=e^{-2\pi i\alpha }$ wherever $f$ is nonzero. Reversing the definition of the jump reverses its sign. Choosing a different branch multiplies both limits by the corresponding common phase.

The factor $z^{-1/4}$ has algebraic branch points at zero and infinity. Near zero the leading behaviour is $z^{-1/4}$; with $w=1/z$, the behaviour near infinity is $g(1/w)\sim w^{9/4}$. These are four-sheeted branch points.

The polynomial denominator vanishes at $z=\pm i$, with nonzero derivative $2z$, and the fractional factor is finite and nonzero near either point on any chosen local branch. HenceA single-valued holomorphic branch is obtained by choosing a branch cut joining zero to infinity. Points along the chosen cut mark its discontinuity, rather than additional isolated poles.

For this real integral choose $0<\arg z<2\pi$, so the upper bank of the positive real axis has $z^{1/4}=x^{1/4}>0$. Use a counterclockwise keyhole contour with inner radius $\epsilon$ and outer radius $R$, enclosing $\pm i$. The upper-bank contribution tends to $J$. On the lower bank, $z^{-1/4}=e^{-i\pi /2}{x}^{-1/4}=-i{x}^{-1/4}$, and the reversed orientation makes its contribution $+iJ$.

The large arc is $O(R^{-5/4})$ because $g=O(R^{-9/4})$, while the small arc is $O({\epsilon }^{3/4})$. Both vanish, givingThe residues on this branch areThe residue theorem yieldsThereforeThe branch cut in (a) was arbitrary; here placing it on the integration axis makes the bank contributions directly proportional to the desired positive-real integral.